# Peter

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I am a beginning math grad student studying for a written qualifier exam. All questions I ask here are taken from old exams, and none are "homework" problems, i.e. for credit (unless tagged as such).

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 2d awarded Popular Question May13 asked Conditional Random Variable and Iterated Expectation May11 awarded Popular Question Feb17 comment QuickSort Average Case Analysis: Probability of a CompareIn the QuickSort algo, the probability that $z_i$ and $z_j$ is compared for $j>i$ is $2/(j-i+1)$. The reason is because if we pick an element between $z_i$ and $z_j$ to act as a pivot before we pick either $z_i$ or $z_j$, then $z_i$ and $z_j$ are fed into different recursive calls and never compared. I don't understand why we ignore $z_1$ up to $z_{i-1}$ and $z_{j+1}$ up to $z_{n}$ in the analysis, where $n$ is size of array. Feb17 asked QuickSort Average Case Analysis: Probability of a Compare Feb10 awarded Yearling Jan4 accepted Closed form for coefficients in Multiple Regression model Jan1 revised Closed form for coefficients in Multiple Regression modeldeleted 1 characters in body Jan1 comment Closed form for coefficients in Multiple Regression modelI've edited my post to make it somewhat clearer. If we think of the predictor variables as random variables $X_i$, I'm asking if the regression coefficients take the given forms. If we had a data matrix of observed values for the explanatory and response variables, then covariance and variance would be replaced by their estimators. Jan1 revised Closed form for coefficients in Multiple Regression modeladded 37 characters in body Jan1 comment Closed form for coefficients in Multiple Regression modelI'm familiar with the normal equation and matrix solution. I was looking for a clean expression for $\beta_i$ that wouldn't require computing a matrix inverse. Is my conjecture for $\beta_i$ incorrect? Jan1 asked Closed form for coefficients in Multiple Regression model Dec30 accepted Integral of $\cos^{n}(x)$ on $[0,2\pi]$ for $n$ a positive integer Dec30 accepted $\epsilon-\delta$ continuity of $1/\sqrt{x}$ Dec30 accepted Norm inequality on Complex Numbers. Dec30 accepted Application of Lagrange Multiplier? Dec28 comment Regressing $Y$ back on the residualsI'm still having some trouble. I understand the orthogonality property, but the equations I am finding doesn't seem correct. Dec27 asked Regressing $Y$ back on the residuals Dec27 accepted What is $Cov(\hat{Y},Y)$? Dec27 awarded Commentator